Comment by pdonis

> he first assertion, that GR is not the whole story, is more or less a given knowing that GR returns non-physical infinities when trying to describe what's inside the BH.

Only at the singularity, but the singularity itself is not even part of the spacetime manifold in GR.

In cases like black holes, there are physical invariants that do increase without bound as the singularity is approached, i.e., still within the spacetime, but their values are still finite at every point within the spacetime.

It is true that most physicists believe that when those invariants reach some particular scale, such as the Planck scale, the GR description in terms of spacetime geometry will break down. But that scale is about twenty orders of magnitude away from what we can currently probe observationally, so this is another of those speculations that, however plausible it seems, is not going to be testable any time soon.